%% DIFFERENTIAL EQUATIONS WITH MATLAB
% CHAPTER 5

%% 
% *An illustration of stability*

%%
% We look at how the solution to
%%
% 
% $$y' = e^{\mbox{--}x} \,\mbox{--} \,2y,\quad y(0)=c$$
% 
% depends on the initial value c.

syms x
eqn = 'Dy = exp(-x) - 2*y'
Y = dsolve(eqn, 'y(0) = c', 'x')

%%
% We plot the solution for initial values .9, 1 and 1.1.

figure;
hold on
for j = -1:1 
    ezplot(subs(Y,'c',1 + j/10),[-1,1])
end
axis tight
hold off

%%
% The inital values are close together.  (Two consecutive functions have
% values which differ by 0.1.)  The plots remain close together for
% positive x, and in fact become closer together. This equation is stable.
% The derivative of the right side with respect to y is negative.

%%
% *An illustration of instability*

%%
% Now we look at the same question for the equation
%%
% 
% $$y' = e^{\mbox{-}x} + 2y,\quad y(0) = c$$
% 

eqn2 = 'Dy = exp(-x) + 2*y'
Y_2 = dsolve(eqn2,'y(0)=c','x')

%%
% We plot the solution for inital values -11/30, -10/30 and -9/30.

figure;
hold on
for j = -1:1 
    ezplot(subs(Y_2,'c',-1/3 + j/30),[-1,1])
end
axis tight
hold off

%%
% The functions  have initial values at x=0 which are close together.     
% (Two consecutive functions have initial values which differ by 1/30).  
% However, the plots spread apart for positve x.  This equation is
% unstable. The derivative of the right side with respect to y is positive.